Stochastic Neural Networks

Hopfield network

A markov chain of neurons, settles to some final state.

• Only two states (1 or -1 / black or white)

• Connections both way with the same weight

  • w_ij = w_ji

• No neurons connect to themselves

  • w_ii = 0

• Asynchronous operation

• sign(v) is the activation function

Can converge very quickly on a custom circuit (but slow on CPU)

• Will fall into some stable state (attractors)

Associative memory

• So given a prompt of part of a dog it will fall back into a state of a reconstructed dog

Learning

Hebbian rule

• w_ij = w_ij + u_ni x_nj

Capacity

• 0.138 U patterns for U neurons... (correct patterns)

• Essentially because it there is too much similarity because so many patterns will start with 110 it is difficult to get the correct one

Will learn spurious states

• Combined states of learned states or kinda random states

Slow in training and slow in evaluation (will also have to settle to the state

Boltzmann machine

A stochastic hidden Hopfield network with hidden units

Activation function is a sidmoid (between 0 and 1)

• Probability of neuron being switched on / off

• T is temperature - how gradual the change between states is

Could have a neuron for input and a neuron for output as the visible units

Anneal the T value, if you in theory cool the system at infinitely slow rate you get the absolute minimia (but you cant really do that)

• Design a schedule for this (high T is stochastic, low T is deterministic)

• Very slow to settle

Learning

• Positive phase

  • ...

• Negative phase

  • ...

Takes forever to settle (with simulated annealing)

Restricted boltzmann machine

(i.e. all the non-connections are forced to be 0, you loose a lot of dynamics)

• No connections between visible units

• No connections between hidden units

• Feedforward

Dynamics

• Easier / faster to train

• Not as powerful

• Autoencoder

  • Lots -> little -> lots (A compression exercise, learn the important ascepts)

Don't need to know the learning with contrastive divergence stuff

Deep Belief Network

• Contrastive divergence

• Stacked auto-encoders

• Fine tuned backprop

• Explaining away labels

First do unsupervised learning, then use those features to derive labels

• Essentially an autoencoder learning step by step, where the 'input' layer moves through the network.

  • A series of restricted boltzmann machines.

  • Requires sigmoids (can't use ReLU)

  • Showed deeper doesn't make it worse

• Then you add the output neurons, and then use backpropagation.

  • Essentially a way of fine-tuning. This was better than SVM.

• Can be used to run computation backwards, to generate possible 'inputs'

Modern Hopfield Network

Can learn more patterns than there are neurons

Kinda like a ReLU unit with some power ^ k

• Ali covers these in better detail in Ali's notes

May be useful for reinforcement applications due to its high capacity!